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Hom-Akivis algebras. (English) Zbl 1249.17005

A Hom-Lie algebra is a Lie algebra with a linear endomorphism \(\alpha \) satisfying \[ [[x,y],\alpha (z)]+[[y,z],\alpha (x)]+[[z,x],\alpha (y)]=0. \] Since Hom-Lie algebras were introduced in [J. Hartwig, D. Larsson and S. Silvestrov, J. Algebra 295, No. 2, 314–361 (2006; Zbl 1138.17012)] this notion has been generalized to other classes of algebraic structures.
An Akivis algebra is an algebraic structure that formalizes the relationship between the commutators and the associators in a non-associative algebra; it has one bilinear antisymmetric operation \([\cdot ,\cdot ]\) and one trilinear operation \((\cdot ,\cdot ,\cdot )\) satisfying the relation \[ [[x,y],z]+[[y,z],x]+[[z,x],y]=(a,b,c)+(b,c,a)+(c,a,b)- (a,c,b)-(c,b,a)-(b,a,c). \] These algebras were first studied by M. A. Akivis [Sib. Math. J. 17, 3–8 (1976); translation from Sibir. Mat. Zh. 17, 5–11 (1976; Zbl 0337.53018)] under the name \(W\)-algebras. Later K. H. Hofmann and K. Strambach [Quasigroups and loops: theory and applications, Sigma Ser. Pure Math. 8, 205–262 (1990; Zbl 0747.22004)] called them Akivis algebras. An important subclass of the class of Akivis algebras are the Maltsev algebras, the tangent algebras of smooth local Moufang loops [A. I. Mal’tsev, Mat. Sb., N. Ser. 36(78), 569–576 (1955; Zbl 0065.00702)]. More generally, the tangent space at the identity of any local loop has the structure of an Akivis algebra, though the correct notion of a tangent structure to a general local loop is that of a Sabinin algebra (for the relation between Sabinin algebras and Akivis algebras, see [I. P. Shestakov and U. U. Umirbaev, J. Algebra 250, No. 2, 533–548 (2002; Zbl 0993.17002)]).
In the present paper the author defines Hom-Akivis algebras, gives low dimensional examples and discusses the influence of special properties of the mapping \(\alpha \). Finally, he considers Hom-Maltsev algebras, Hom-flexible algebras and Hom-alternative algebras.

MSC:

17A30 Nonassociative algebras satisfying other identities
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17D10 Mal’tsev rings and algebras
17D05 Alternative rings
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